The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Cauchy’s approach was a significant logical improvement over the infinitesimal approach of Leibniz because, instead of invoking the metaphysical notion of infinitesimals, the quantities dy and dx annales Scientifiques de L’Ecole Normale Superieure, Volume 14 PDF now be manipulated in exactly the same manner as any other real quantities in a meaningful way. In physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails.
Following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. The differential is defined in modern treatments of differential calculus as follows. One or both of the arguments may be suppressed, i. This notion of differential is broadly applicable when a linear approximation to a function is sought, in which the value of the increment Δx is small enough. 1 is the principal part of the change in y resulting from a change dx1 in that one variable. This is because the derivative fx with respect to the particular parameter x gives the sensitivity of the function f to a change in x, in particular the error Δx.
As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. This additional factor tends to make the error smaller, as ln b is not as large as a bare b. When the independent variable x itself is permitted to depend on other variables, then the expression becomes more complicated, as it must include also higher order differentials in x itself. Similar considerations apply to defining higher order differentials of functions of several variables. In more variables, an analogous expression holds, but with an appropriate multinomial expansion rather than binomial expansion.
Higher order differentials in several variables also become more complicated when the independent variables are themselves allowed to depend on other variables. A mon avis, rien du tout. That is: Finally, what is meant, or represented, by the equality ? In my opinion, nothing at all.
Then the nth differential defined in this way is a homogeneous function of degree n in the vector increment Δx. The higher order Gateaux derivative generalizes these considerations to infinite dimensional spaces. A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. An operation d with these two properties is known in abstract algebra as a derivation. Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantity dt. More general analogous expressions hold, in which the intermediate variables x i depend on more than one variable. Rn be a pair of Euclidean vectors.
0, then f is by definition differentiable at the point x. If t represents time and x position, then h represents a velocity instead of a displacement as we have heretofore regarded it. This yields yet another refinement of the notion of differential: that it should be a linear function of a kinematic velocity. Defining the differential as a kind of differential form, specifically the exterior derivative of a function. The infinitesimal increments are then identified with vectors in the tangent space at a point. Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry.
Differentials in smooth models of set theory. Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers which contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson. For a detailed historical account of the differential, see Boyer 1959, especially page 275 for Cauchy’s contribution on the subject.